🧄Calculating the Area of a Half-Sphere Using Cylindrical Coordinates

Calculating the surface area of a parameterized surface like a half-sphere involves integrating the magnitude of the cross product of its partial derivative vectors, which is equivalent to the square root of the determinant of the induced metric tensor $( \left.g_{\alpha \beta}\right)$, where $g_{\alpha \beta}$ components are the dot products of the partial derivatives with respect to the parameters, and in this case, a cylindrical parametrization simplifies the integral by defining infinitesimal area elements ( $d S=\frac{R \rho}{\sqrt{R^2-\rho^2}} d \rho d \phi$ ) and allowing a u-substitution ( $u = R ^2- \rho ^2$ ) to yield the final result of $2 \pi R ^2$, confirming half the area of a full sphere.

🎬visualizes the surface area calculation of a half-sphere using the provided cylindrical parametrization

The surface area of a complex 3D shape, like a half-sphere, can be precisely calculated by breaking it down into infinitesimally small patches. The app shows how the cylindrical parametrization of a sphere allows us to define these tiny patches in terms of $\rho$ and $\phi$ and then sum them up with integration to find the total area. It visually demonstrates that the total area is a cumulative sum of all these smaller surface elements, which change shape and size as you move across the sphere's surface.

🖊️Mathematical Proof

Calculating the Area of a Half-Sphere Using Cylindrical Coordinates | Proof and Derivationviadean on Notion